#329761

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,16,13,17,12,18)(11,15,21,22,19,24,20,23)(25,26,29,33,30,28,31,34)(27,35,39,36,38,32,37,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,22)(17,24)(18,23)(25,27)(26,32)(28,35)(29,39)(30,38)(31,37)(33,40)(34,36);(1,3)(2,6)(4,7)(5,8)(10,16)(12,14)(15,22)(17,18)(20,21)(23,24)(25,28,30,26)(27,36,38,40)(29,33,31,34)(32,39,35,37);(3,7)(6,8)(10,17)(11,19)(16,18)(20,21)(25,29,30,31)(26,33,28,34)(27,37,38,39)(32,36,35,40)

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236957);